For finite-volume hyperbolic 3-manifolds, the hyperbolic metric uniquely minimizes minimal surface entropy among sectional curvature at most -1 metrics and uniquely maximizes it among scalar curvature at least -6 metrics under rigidity or C0-closeness hypotheses.
Conformal currents and the entropy of negatively curved three-manifolds
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abstract
In this paper, we describe the intersection between geodesic and conformal currents on closed hyperbolic three-manifolds. We use this to prove some sharp bounds which involve the Liouville entropy of a negatively curved metric, the minimal surface entropy, and the area ratio. Using these ideas we also give a new proof of the Mostow Rigidity Theorem in the three-dimensional case.
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Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds
For finite-volume hyperbolic 3-manifolds, the hyperbolic metric uniquely minimizes minimal surface entropy among sectional curvature at most -1 metrics and uniquely maximizes it among scalar curvature at least -6 metrics under rigidity or C0-closeness hypotheses.